8th Class Mathematics Mathematical Operations Mathematical Operations

Mathematical Operations

Category : 8th Class

Introduction

Mathematical Operations mean things like add, subtract, multiply, divide, squaring, etc. It isn’t a number it is probably an operation.

 

MATHEMATICAL OPERATIONS

This section deals with questions on simply mathematical operations. There are four fundamental operations. There are four fundamental operations, namely:

Additions i.e, + ; Subtraction i.e,-;

Multiplication i.e. X; and Division i.e., \[(6\div 2)\times 3=0\]

There are also statements such as Less than i.e. <, greater than i.e. >, and equal to i. e = , not equal to i. e \[\div and\times ,2and3\],  etc.

>, and equal to i. e=, not equal i.e \[\times to-,2 and 6\], etc.

Such operations are represented by symbols different from the usual ones. The questions involving these operations are coded using artificial symbols. The candidate has to make a substitution of the real signs and solve the equation accordingly.

We always, while solving a mathematical expression, proceed according to the rule B 0 D M A S.

i.e, B for Brackets; O for' of (literally multiplication),

D for division; M for multiplication, A for additions and S for subtraction in sequence.

 

DIFFERENT TYPES OF PROBLEMS

 

TYPE - I

 

Problem-solving By Substitution

In this type, you are provided with substitutes for various mathematical symbols or numbers. Followed by a question involving calculation of an expression or choosing the correct/ incorrect equations. The candidate is required to put in the real signs or numerals in the given equation and then solve the questions as required.

 

EXAMPLE 1:

If L stands for +, M stands for-, N stands for x, P stands for -,then 1 4 N 10 L 42 P 2 M 8 =?

(a) 153                 (b) 216

(c) 248                   (d) 251

Sol.        (l) Using the proper signs, we get Given expression.

                \[\div and\times ,2and6\]

                \[\times to-,2and3\]

 

DIRECTIONS (Example 2-4): In each of the following examples which one of the four interchanges in signs and numbers would make the given equation correct?

 

EXAMPLE 2:

                \[(6\div 3)-2=0\]

(a) \[2-2=0\]      (b) \[0=0,\]

(c) \[a\cancel{<}b\]         (d) None of these

Sol.        (c) On interchanging + and 4 and 6, we get the equation as

\[a\ne b\], or \[a\cancel{>}b\] or \[a\ne b\] which is true

 

EXAMPLE 3:

\[a\cancel{>}b\]

(a) \[a\cancel{<}b\]       (b)  \[\phi =\]

(c) \[\Delta =\]                  (d) \[p\square qOr,\]

Sol.        (c) On changing - to + and interchanging 2 and 6, we get the equation as

\[p\phi q\square r\] or \[p\phi q\times r\] or \[p+q\times r\] which is true.

 

EXAMPLE 4:

                \[p\Delta q\phi r\]

(a) \[\Rightarrow \]         (b) \[\Delta \]

(c) \[\phi \]         (d) \[\Rightarrow \]

Sol.        (d) One changing x to - and interchanging 2 and 3, we –get the equations as

\[\Delta \] or \[p\Delta q\]or \[p\times q\times r\], which is true.'

 

MATHEMATICAL LOGIC

Consider the statement "5 is greater than 3 ".

Now consider which of the following statements are true and which are false.

“5 is not greater than 3”                (False)

“5 is equal to 3”                                 (False)

“5 is less than 3”                               (False)

“5 is not equal to 3”                         (True)

“5 is not less than 3”                       (True)

In general, between any two numbers a and b, only one of the following relations can exist at a time

a > b

or            a < b

or            a = b

If             a > b, then \[p\phi q\times r\] and \[p+q\times r\]

If             a < b, then \[p\Delta q\phi r\]b and \[\Delta \]

If             a = b, then \[\Delta \] and \[\Rightarrow \]  

DIRECTIONS (Example 5-6): Let the following symbols denote some relationships between numbers.

O = greater than                               \[\Delta \]not greater than

+ = equal to                                        \[\Rightarrow \] not equal to

= less than                                          x = not less than

 

EXAMPLE 5:

If \[\square \] O r, it is possible that

(a) \[\Delta \]                    (b) \[\Rightarrow \]

(c)  \[\phi \]                        (d) \[\Delta \]

Sol.        (b)  D (less than \[\Rightarrow \] \[\Rightarrow \](Not equal to or \[\Delta \] (not greater than)

O (greater than ) \[\Rightarrow \]\[p\Delta q\] (Not equal to) or x (not less than)

For p, and q, option (a), (b), (d) are possible.

For q and r, options (b), (c) are possible.

Hence the answer is (b)

 

EXAMPLE 6:

If \[\Rightarrow \] O r, it is possible that

(a) \[p\square p\]                            (b) \[p\phi q\]

(c) \[p\times q\]               (d)  \[\Rightarrow \]

Sol.        (a) \[q\Delta r\] = note equal to

Hence, \[q\times r\]\[16-8\div 4+5\times 2=8\] O (greater than)

or \[\div \]\[\times \]\[-\] (Less than)

\[\div \]\[-\]\[+\] (not greater than)

or  \[-\]\[+\]x (not less than)

Similarly, O \[=16\div 8-4+5\times 2=2=2-4+10=8\]\[56\div 7\times 2+8-1=9\] and O \[\times \]x

hence \[-\]\[-\] or p  O p or \[+\] or \[-\]or \[+\]

and    q O r \[-\] \[=56\div 7\times 2-8+1=16-8+1=9\] or \[121\div 11-3\times 13+2=22\]

All four options are possible so far as p and q are concerned.

Between q and r, only the first is correct.

 

TYPE - II

 

DIRECTIONS (Example 7- 9): In each of the following questions, an equation becomes incorrect due to the interchange of two signs. One of the four alternatives, specifies the interchange of sign in the equations, which when made, will make the equation correct. Find the correct alternative.

 

EXAMPLE 7:

\[-\]

(a) \[\times \] and \[-\]                

(b) \[\div \]         and \[\div \]

(c) \[-\]and \[+\]             

(d) \[-\] and \[=121\div 11\times 3-13+2\]

Sol.        (b) On interchanging - and - we get: Given expression \[=\frac{121}{11}\times 3-13+2=11\times 3-13+2=22\]

 

EXAMPLE 8:

                \[32\times 41=15;51\times 34=47;41\times 52=37,\]

(a)  \[87\times 53=?\]and \[32\times 41=(3-2)(4+1)=15;\]             (b) \[51\times 34=(5-1)(3+4)=47\] and x

(c) \[\therefore \] and \[87\times 53=(8-7)(5+3)=18\]                    (d) \[{{7}^{*}}1=64:\]and \[{{3}^{*}}9=144\]

Sol.        (c) On interchanging-and +, we get

Given expression \[{{5}^{*}}6\]

 

EXAMPLE 9:

                \[a*b={{(a+b)}^{2}},\]

(a) \[\therefore \] and \[5*6={{(5+6)}^{2}}=121.\]                           (b) \[2\times 1=81;3\times 2=278;2\times 5=8125,\]and \[1\times 3=\]

(c) \[a\times b={{a}^{3}}{{b}^{3}}\] and \[2\times 1={{2}^{3}}{{1}^{3}}=81\]         (d) \[1\times 3={{1}^{3}}{{3}^{3}}=127\] and \[2\div 3=89;3\div 4=2716;4\div 3=649,\]

Sol.        (a) On interchanging + and x we get

Given expression \[1\div 2=?\]

\[a\div b={{a}^{3}}{{b}^{2}},\]

 

TYPE - III

 

DIRECTION (Example 10): In the following question, three statements of numbers following same rules are given. Find the rule and accordingly find the value of the number?

 

EXAMPLE 10:

If \[\therefore \]then \[1\div 2={{1}^{3}}{{2}^{2}}=14.\]

(a) 68                                     (b) 64

(c) 85                                     (d) 18

Sol.        (d) The logic is \[\div \]

\[\div \]etc.

\[15-5\div 5\times 20+10=6\] \[8\div 10-3+5\times 6=8\]

 

EXAMPLE 11:

If \[6\times 2+3\div 12-3=15\]

\[3\div 7-5\times 10+3=10\]

What is the value of \[=15-5\div 5\times 20+10\]?

(a) 22                                     (b) 55

(c) 66                                     (d) 121

Sol.        (d) Rule is \[=15\times 5+5-2=75+5-2=78\]

\[=8+10\times 3\div 5-6=8+10\times \frac{3}{5}-6=8+6-6=8.\] \[=6-2\div 3+12\times 3=6-\frac{2}{3}+36=42-\frac{2}{3}=\frac{124}{3}\]

 

TYPE - IV

DIRECTIONS (Example 24 - 25): In each of the following questions, three statements of numbers following same rules are given. Find the rule and accordingly find the value of the number.

 

 EXAMPLE 12:

If \[=3+7\times 5-10\div 3=3+7\times 5-\frac{10}{3}=\frac{104}{3}\], then \[\therefore \]

(a) 127                                  (b) 271

(c) 126                                   (d) 129

Sol.        (a) The rule is \[2-5+3=4\]

\[3-2+5=4\]etc. So, \[6=4,\]

 

EXAMPLE 13:

If \[5-3+2=4,\]then \[+\]

(a) 21                                     (b) 42

(c) 14                                     (d) 81

Sol.        (c) The rule is \[\div \]

\[-\] \[\div \]

 

Miscellaneous Solved Examples

 

EXAMPLE 1:

If x stands for \[-\], stands for +, + stands for \[+\] and – stands for x, which one of the following equations is correct ?

(a) \[8+6\times 4\div 3-4=\]

(b) \[-12\]

(c) \[-20/3\]

(d) \[12\]

Sol.        (b) Using the proper signs, we get

Expression in (a) \[20/3\]

\[=8\div 6-4\times 3+4\]

Expression in (b)

                \[=\frac{4}{3}-4\times 3+4\]

Expression in (c)

\[=\frac{4}{3}-12+4=-\frac{20}{3}\]

Expression in (d)

\[=18\times 14+6-16\div 4\]

\[=18\times 14+6-4=254\] Statement (b) is true

 

EXAMPLE 2:

Which one of the four interchanges in signs and numbers would make the given equation correct?

 

(a) + and-, 2 and 3            (b) + and-, 2 and 5

(c) + and-, 3 and 5            (d) None of these

Sol.        (c) By making the interchanges given in (a), we get the equations as \[-\], which is false. By making the interchanges given in (b), we gets the equations as \[\div \] or \[\frac{5}{9}\], which is false. By making the interchanges given in (c), we get the equations as \[=15\div 3+24-12\times 2\], which is true.

So, the answer is (c)

 

EXAMPLE 3:

If \[=5+24-24=5\] means \[\text{3  }\!\!\times\!\!\text{  2  4 O 6 + 3  2}\], x means \[\text{3 + 2  4 O 6  3 }\times \text{ 2}\],\[\text{3  2  4 - 6 }\times \text{ 3 }\times \text{ 2}\] means x and \[\text{3 }\times \text{ 2 }\times \text{ 4 = 6 + 3  2}\] means \[3+2-4>6x3+2\], then \[3\div 2-4>6x3+2\]

(a) \[3\times 2-4=6+3+2\]                                            (b) \[3+2+4<6\div 3-2\]

(c) \[-\]                                 (d) \[\div \]

Sol.        (b) Given expression \[+\]

                \[+\]

\[\div \]

 

EXAMPLE 4:

If a means' Plus', b means minus', c means' multiplied by and d means' divides by1, then 18 c 14 a 6 b 16 d4 = ?

(a) 63                                     (b) 254

(c) 288                                   (d) 1208

Sol.        (b) Given expression \[-\]

                                                \[~15-5\div 5\times 20+10=6\]

 

EXAMPLE 5:       

If A means  \[8\div 10-3+5\times 6=8\], B means \[6\times 2+3\div 12-3==15\], C means + and D means x, the 15 B 3 C 24 A 12 D2=?

(a) 34                                     (b) 2

(c) \[3\div 7-5\times 10+3=10\]                                 (d) 5

Sol.        (d) Given expression \[=15\times 5+5-20+10\]

                                                \[=15\times 5+5-2=75+5-2=78\]

 

EXAMPLE 6:

If x stands for addition, < stands for subtraction, > stands for multiplication, + stands for division, 0 stands for greater than, = stands for less than and - stands for equal to then-

(a) \[=8+10\times 3\div 5-6\] 

(b) \[=8+10\times 3\times \frac{1}{5}-6=8+6-6=8\]

(c) \[(3\div 4)+2=2\]  

(d) \[+\]

Sol.        (a) We may rewrite all the statements using the meaning of the symbols used. We therefore have

(a) \[+\] (True)

We may check the other statements also to make sure that our answer is correct        

(b) \[+\] (False)

(c) \[(10C4)+\](False)

(d) \[(4C4)\] (False)

 

EXAMPLE 7:

If x stands for\[84\oplus 72=45;\], \[63\oplus 41=33,\] stands for \[25\oplus 52=33,\], \[94\oplus 82=?\] stands for \[5\times 9=144;\] and  \[7\times 8=151:\] stands for x, which one of the following equations is correct?

(a)\[\div \]

(b) \[\times \]

(c) \[+\]

(d) \[-\]

Sol.        (b) Using proper sign, we have

(a)  \[+\]

\[\div \](False)

(b) \[-\]

\[\times \](True)

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